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Petrov-Galerkin operator inference with application to stability-encouraging identification
Johannes Rettberg, Jonas Nicodemus, Harsh Sharma, Boris Kramer, Jörg Fehr, Benjamin Unger
TL;DR
Standard operator inference uses Galerkin projections, while the connection between Petrov–Galerkin projection and OpInf remains largely unexplored. This paper extends OpInf to prescribed Petrov–Galerkin projections, analyzes their relation to intrusive reduced operators, and develops pH identification approaches. In the poroelasticity example, 68% of PG-OpInf models were stable, compared with 6% of G-OpInf models.
Problem
Standard OpInf typically uses Galerkin projection, and its connection to Petrov–Galerkin reduction—which may preserve system properties such as stability and passivity—remains largely unexplored.
Source: §1 · p. 1, §2.2 · p. 7Method
The paper extends OpInf to prescribed Petrov–Galerkin projections, analyzes the relation between inferred and intrusive operators, and formulates convex pH identification.
Source: §1 · p. 2, §3.2.3 · p. 15Results
For poroelasticity, 68% of PG-OpInf models were stable, compared with 6% of G-OpInf models.
Source: §4.2 · p. 18Takeaways & Limitations
PG-OpInf is stability encouraging with a pH-motivated test basis, but does not guarantee stability of the resulting reduced models.
Source: §4.2 · p. 18, §5 · p. 20When the Hamiltonian Hessian is unknown, the current approach for determining the test basis is not yet fully satisfactory.
Source: §4.3 · p. 19
Abstract
from arXiv · showhide
Data-driven model order reduction methods such as operator inference enable the efficient construction of reduced-order models directly from high-dimensional time-domain data. Standard operator inference typically seeks a Galerkin-type reduced model in a prescribed low-dimensional subspace by identifying its reduced operators from projected snapshot data. The resulting inference problem is formulated as a least-squares problem admitting an efficient closed-form solution. However, it is well known from intrusive model order reduction for linear time-invariant systems that Petrov-Galerkin projections can additionally preserve important system properties such as stability and passivity. To overcome the limitations of standard operator inference, we extend the framework for linear time-invariant systems to incorporate Petrov-Galerkin projections and provide explicit error expressions and bounds between the intrusive and nonintrusive reduced operators, thus generalizing results from the literature. We demonstrate the proposed approach in the context of dissipative and port-Hamiltonian systems. Furthermore, we introduce a novel convex optimization formulation that explicitly enforces the port-Hamiltonian structure on the inferred operators. The effectiveness of the proposed methods is demonstrated on several well-established benchmark problems, including the CD player, an atmospheric model, a mass-spring-damper system, and a poroelasticity system.
1. Introduction
The paper extends nonintrusive operator inference to prescribed Petrov–Galerkin projections, connecting data-driven reduced operators to intrusive reduction. It develops theory and port-Hamiltonian identification approaches, motivated by the gap between standard Galerkin OpInf and the role of Petrov–Galerkin methods in structure-preserving reduction.
1.1. Contributions.
- The framework characterizes nonintrusive Petrov–Galerkin OpInf operators as intrusive operators plus a correction for unresolved state components, and derives an explicit error bound.The analysis also finds that vanishing unresolved components alone need not ensure convergence, as the least-squares problem may become more ill-conditioned.Source: §1.1 · p. 2, §1 · p. 2
- A pH-motivated Petrov–Galerkin test basis promotes stability in inferred models, while estimating the Hamiltonian matrix from sampled Hamiltonian data addresses the need to know it.Constructing the test space requires the quadratic Hamiltonian.Source: §1.1 · p. 2
- The one-shot pH identification approach fits a low-dimensional linear system from state, input, and output data through convex optimization.It avoids requiring the Hamiltonian, nonlinear optimization, or fixing one matrix while optimizing another.Source: §1.1 · p. 3
1.2. Literature Review and State of the Art.
- The literature review situates OpInf among reduced-operator identification methods, describing its use of state-derivative data and extensions to parameter-dependent, output, and nonlinear polynomial models.It also contrasts OpInf with DMD, which seeks a spectral decomposition that continues the state from one snapshot to the next.Source: §1.2.1 · p. 3
1.2.2. Structure-preserving Model Order Reduction of Port-Hamiltonian Systems.
- The review describes structure-preserving intrusive pH reduction across interpolation, moment-matching, balanced truncation, Dirac-structure, and optimization-based approaches, then focuses on nonintrusive methods.It frames pH identification as incorporating physical knowledge, with methods differing by domain, data requirements, and pH-system type.Source: §1.2.2 · pp. 3–4, §1.2.3 · p. 4
1.2.3. System Identification of Port-Hamiltonian Systems.
- The pH-identification review covers linear methods using frequency-domain data, time-domain transformations, optimization, reparameterization, and pH DMD; the latter requires the Hamiltonian Hessian.It also notes fewer nonlinear pH identification works, often under restrictive assumptions.Source: §1.2.3 · p. 4
1.3. Organization of the Manuscript.
- The manuscript proceeds from pH and MOR fundamentals to Petrov–Galerkin OpInf, pH extensions, numerical examples, and conclusions.The pH extensions include approaches for deriving the Hamiltonian Hessian and a convex formulation.Source: §1.3 · pp. 4–5
1.4. Notation.
- The notation section defines matrix-set symbols and uses A⊺ and ∥A∥F for matrix transpose and Frobenius norm.It also specifies that A ≻ 0 and A ⪰ 0 denote symmetric positive definite and positive semidefinite matrices.Source: §1.4 · p. 5
2. Preliminaries
The preliminaries define the LTI models and stability-related structures considered, then outline intrusive projection-based reduction and the data-driven OpInf workflow. They distinguish Petrov–Galerkin from Galerkin projection and note that arbitrary projections need not preserve stability or passivity.
2. Preliminaries
- The LTI full-order model is specified by its state, input, output, and system matrices, with a transfer-function representation in the Laplace domain.The model is considered over a finite time interval, and its state dimension may be large relative to its input and output dimensions.Source: §2 · pp. 5–6
2.1. Dissipative-Hamiltonian and Port-Hamiltonian Systems.
- Dissipative-Hamiltonian structure characterizes asymptotic stability, while port-Hamiltonian structural constraints imply passivity and stability.The stated stability condition requires all eigenvalues to lie strictly in the left half-plane for asymptotic stability.Source: §2.1 · p. 6
2.2. Intrusive Model Order Reduction.
- Intrusive Petrov–Galerkin reduction uses trial and test spaces to form reduced operators, but an arbitrary such projection need not preserve stability or passivity.The reduced descriptor matrix is WᵀV; setting W=QV gives a port-Hamiltonian reduced model with structure-preserving reduced operators.Source: §2.2 · p. 7
- For the port-Hamiltonian projection, the reduced matrices retain symmetry and skew-symmetry properties, and the reduced model is itself port-Hamiltonian.The descriptor matrix is symmetric positive definite, the reduced interconnection matrix is skew-symmetric, and the dissipation matrix is symmetric positive semidefinite.Source: §2.2 · p. 7
2.3. Operator Inference.
- Operator inference uses state, derivative, input, and output snapshots, projects the data onto a reduced basis, then fits reduced operators by least squares.The minimum-norm solution uses a Moore–Penrose pseudoinverse and is unique exactly when the data matrix has full row rank.Source: §2.3 · pp. 7–8
3. Methodology
The methodology extends operator inference to prescribed Petrov–Galerkin spaces and analyzes how its inferred operators relate to intrusive projections. For port-Hamiltonian systems, it develops routes to infer reduced models, including a convex formulation that enforces structure.
3. Methodology
- Petrov–Galerkin OpInf uses a structure-preserving projection in the nonintrusive framework while retaining a least-squares inference problem.The test basis is assumed given, and the projected problem can be solved with a pseudoinverse.Source: §3 · p. 8, §3.1 · p. 9
3.1. Linear Petrov–Galerkin Operator Inference
- The data are projected obliquely using a prescribed test basis, and the resulting Petrov–Galerkin inference problem is solved by pseudoinverse.The setup allows for discretization or measurement errors.Source: §3.1 · pp. 8–9
3.1.1. Error Analysis of Petrov–Galerkin Operator Inference
- OpInf operators equal intrusive Petrov–Galerkin operators plus corrections for unresolved states and time-discretization errors; under certain reprojection conditions, the corrections vanish.When they vanish, the inferred and intrusive operators coincide.Source: §3.1.1 · pp. 9–10
3.1.2. Convergence Analysis of Petrov–Galerkin Operator Inference
- For nonzero inputs, convergence of the unresolved-state error term is not guaranteed; the analysis gives a stronger result for autonomous systems under its rank assumption.The explicit error bound is presented for a single input, while analogous multiple-input bounds are expected to require additional range assumptions.Source: §3.1.2 · pp. 10, 12
3.2. Port-Hamiltonian Petrov–Galerkin Operator Inference
- The general port-Hamiltonian inference problem is constrained, nonlinear, and nonconvex, motivating complementary routes that either use Hamiltonian information or avoid requiring it.The routes include Petrov–Galerkin inference with known Hamiltonian Hessian, data-based Hessian estimation, and a one-shot convex formulation.Source: §3.2 · p. 12
3.2.1. Port-Hamiltonian Petrov–Galerkin Operator Inference
- With known Hamiltonian Hessian Q, the method uses test basis W = QV and identified operators converge under convergent time integration and full-column-rank data.The residual is defined using the port-Hamiltonian projector.Source: §3.2 · p. 13, §3.2.1 · p. 13
3.2.2. Inferring the Hessian of the Hamiltonian
- When Hamiltonian samples are available, the Hessian can be estimated from them; in the small-data regime, a positive-definiteness constraint is imposed through semidefinite programming.The unconstrained least-squares estimate may be indefinite, and post-hoc projection can poorly approximate the true Hessian.Source: §3.2.2 · pp. 13–14
3.2.3. Port-Hamiltonian Inference Through Convex Optimization
- A one-shot convex formulation directly incorporates port-Hamiltonian matrix constraints without requiring prior knowledge of Q.The optimization is a semidefinite program; the method relies on a standard projection strategy and POD-based basis in this setting.Source: §3.2.3 · pp. 14–15
4. Numerical Experiments
The experiments evaluate operator recovery, stability and accuracy, and Hamiltonian-inference strategies across four benchmark systems. Petrov–Galerkin methods generally improve stability and error, but do not guarantee stable ROMs.
4. Numerical Experiments
- The experiments use four benchmarks and compare intrusive G-POD and PG-POD with G-OpInf, PG-OpInf, and two unknown-Hessian approaches.The benchmarks are the CD player, earth atmosphere, mass-spring-damper, and poroelasticity systems.Source: §4 · p. 15
4.1. Distance to Intrusive Operators and Error Bounds.
- Across reduced dimensions and examples, εref equaled εcorr, but had no convergence rate despite decaying σ(X); εest and εbound remained nearly constant.Figure 1 displays the theorem-related quantities and the singular values of the state data matrix.Source: §4.1 · p. 16, Figure 1 · p. 17
4.2. Stability-encouraging Property of PG-OpInf.
- PG-OpInf ROMs were stable at every tested mass-spring-damper order and unstable at only five reported poroelasticity orders; Petrov–Galerkin methods generally had lower H∞ errors than Galerkin methods.The comparison covers reduced orders 2, 8, …, 80; Figure 2 marks stable-component errors with dashed lines when the original ROM is unstable.Source: Figure 2 · p. 18, §4.2 · p. 18
4.3. Inferring the Hamiltonian and the pH One-shot Identification.
- In the original sampling setting, estimating the Hamiltonian Hessian failed and yielded large-magnitude entries (~10^6), while sampling every third time point produced a reasonable estimate.The increased-data setting corresponds to a sampling interval of kδt = 1.5.Source: §4.3 · p. 19
- With unknown Q, the current method for choosing W is not yet satisfactory; improving Q estimation or finding alternative choices of W remains future work.The authors identify alternative test-space choices as an active area of investigation.Source: §4.3 · pp. 19–20
- All methods produced stable ROMs at every order; known-Q PG-OpInf had the smallest H∞ errors for most orders, while unknown-Q variants had errors comparable to G-OpInf.Figure 3 compares G-OpInf, PG-OpInf, PG-OpInf-H, and pHOpInf-CVX for the mass-spring-damper example.Source: §4.3 · p. 19, Figure 3 · p. 20
5. Conclusions
The paper develops Petrov–Galerkin operator inference for LTI systems and analyzes its relation to intrusive Petrov–Galerkin reduction. It also presents stability-encouraging pH approaches, including a convex formulation that guarantees pH structure.
5. Conclusions
- The paper characterizes the relation between identified and intrusive Petrov–Galerkin operators and shows they do not generally converge to one another, except in special settings such as constant zero input.The authors validate the theoretical analysis through numerical experiments on benchmark examples.Source: §5 · p. 20
- Numerical examples show PG-OpInf is stability-encouraging when its test basis is chosen using a structure-preserving projection motivated by pH systems.The stated result is that stability is encouraged, not guaranteed.Source: §5 · p. 20
- To address the need for the Hamiltonian Hessian, the paper proposes inferring it from Hamiltonian measurements or using one-shot operator inference with constrained convex optimization that guarantees pH structure.Source: §5 · p. 20